I was recently asked 'how would you implement a hastable'. I know the hashing algorithm is critical as the less collisions the better WRT performance, but what algorithm/data structure should be employed to deliver amortized constant time {O(1)} for insert/delete/lookups?
Hash tables have two main possibilities:
The important part about hash tables [in both solutions] in order to allow armotorized O(1) insert/del/look up - is allocating a bigger table and rehashing once a pre defined load factor was reached.
EDIT: complexity analsis:
Assume a load factor of p
for some p < 1
.
p
Thus the mean of array accesses is: Sigma(i * p^(i-1) * (1-p)) for each i in [1,n]
This gives you: Sigma(i * p^(i-1) * (1-p)) = (1-p) * Sigma(i * p^(i-1)) <= (1-p) * 1/(p-1)^2 = 1-p = CONST
. [have a look at the correctness of Sigma(i * p^(i-1)) < 1/(p-1)^2 in wolfram alpha ]. Thus resulting on average a constant number of accesses to the array. Also: You might need to rehash all elements after the load factor was reached, resulting in O(n)
accesses to the array. This results in n * O(1)
ops [adding n elements] and 1 * O(n)
op [rehashing], so you get: (n * O(1) + 1 * O(n)) / n = O(1)
armotorized time.
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